Further Statistics

Further Statistics extends the statistical methods from A-Level Mathematics, introducing continuous probability distributions, more sophisticated hypothesis tests, and the chi-squared family of tests for goodness of fit and independence.

Topics Covered

Poisson and Geometric Distributions

• Poisson distribution — $X \sim \text{Po}(\lambda)$; derivation as the limit of $\text{Bin}(n, p)$ as $n \to \infty$, $p \to 0$ with $np = \lambda$
• Poisson properties — mean $= \lambda$, variance $= \lambda$; additive property of independent Poissons
• Geometric distribution — $X \sim \text{Geo}(p)$; $P(X = x) = (1-p)^{x-1}p$; memoryless property
• Hypothesis testing — using Poisson and geometric distributions; critical regions, significance levels, $p$-values

Exponential and Continuous Random Variables

• Exponential distribution — $X \sim \text{Exp}(\lambda)$; PDF $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$; CDF $F(x) = 1 - e^{-\lambda x}$
• Link to Poisson processes — the waiting time between Poisson events follows an exponential distribution
• Continuous random variables — PDF, CDF, $E(X) = \int xf(x)\,dx$, $\text{Var}(X) = E(X^2) - [E(X)]^2$
• Median and mode — finding the median from the CDF; locating the mode from the PDF

Chi-Squared Tests

• Goodness of fit — testing whether observed data follows a specified distribution; $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$
• Contingency tables — testing for independence between two categorical variables
• Degrees of freedom — calculating $\nu$ correctly; $\nu = k - 1$ for goodness of fit, $\nu = (r-1)(c-1)$ for contingency tables
• Combining cells — when expected frequencies are below 5
• Interpretation — what a significant result actually means in context

Study Tips

1. Know when to use each distribution — Binomial for fixed trials, Poisson for rare events in a fixed interval, Geometric for “first success” problems, Exponential for continuous waiting times.
2. Practise calculating expected frequencies — for chi-squared tests, the expected values must be calculated correctly before you can compute the test statistic.
3. Show all working in hypothesis tests — state $H_0$ and $H_1$, calculate the test statistic, compare to critical value or find the $p$-value, state the conclusion in context.
4. Understand the memoryless property of both Geometric and Exponential distributions — it is a common exam topic that tests deep understanding.
5. Check integration — continuous random variable problems require careful definite integration. Always verify bounds from the support of the distribution.

Hypothesis Testing Workflow

Every hypothesis test follows the same five-step structure:

1. State hypotheses — $H_0$ (null: no effect/difference) and $H_1$ (alternative)
2. Choose significance level — $\alpha = 0.05$ or $0.01$
3. Calculate the test statistic — using the appropriate distribution
4. Determine the critical region or $p$-value — compare to $\alpha$
5. State the conclusion in context — never just “reject $H_0$”; explain what this means for the real-world situation

Distribution Selection Guide

Scenario	Distribution	Key Parameters
Counting events in a fixed interval	Poisson($\lambda$)	Mean = Variance = $\lambda$
First success in repeated trials	Geometric($p$)	$P(X=x) = (1-p)^{x-1}p$
Continuous waiting time	Exponential($\lambda$)	Mean = $\frac{1}{\lambda}$
Testing goodness of fit	$\chi^2$	Degrees of freedom $\nu$

How to Use These Notes

Follow the sidebar order. Each page provides formal distribution definitions, worked calculation examples, and exam-style hypothesis testing problems. Start with Poisson and Geometric, then move to continuous distributions, then chi-squared tests.